This work reveals an experimental microscopy acquisition scheme successfully combining Compressed Sensing (CS) and digital holography in off-axis and frequency-shifting conditions. CS is a recent data acquisition theory involving signal reconstruction from randomly undersampled measurements, exploiting the fact that most images present some compact structure and redundancy. We propose a genuine CS-based imaging scheme for sparse gradient images, acquiring a diffraction map of the optical field with holographic microscopy and recovering the signal from as little as 7% of random measurements. We report experimental results demonstrating how CS can lead to an elegant and effective way to reconstruct images, opening the door for new microscopy applications.
General high resolution microscopy involves dense data acquisition. One intense field of research aims to reduce the amount of data acquisition or sample illumination [1,2]. In [1], the acquisition is restricted to only those areas where relevant signal is present. In [2] a method called controlled light-exposure microscopy (CLEM) is introduced, supported by a nonuniform illumination of the field of view. However, both methods suffer from being image-content dependent for a successful implementation. Indeed, these methods need a feedback loop inside the acquisition setup to make decisions about the sampling rate or the illumination intensity, depending on the objects characteristics. Here, we address the sensing problem in microscopy by taking an alternative approach provided by the new theoretical framework of Compressed Sensing (CS). This method is independent of image-content and does not need any feedback loop during the acquisition. CS was previously reported in magnetic resonance imaging acquisition [3], single-pixel imaging [4] or inline, single-shot holography for tridimensional imaging [5]. The main idea presented here is to combine off-axis, frequency-shifting (for accurate phase-shifting) digital holography to perform quadrature-resolved random measurements of an optical field in a diffraction plane and a sparsity minimization algorithm to reconstruct the image.
CS is a novel mathematical theory for sampling and reconstructing signals in an efficient way, introduced by Candès and Donoho [6][7][8]. It exploits the fact that most images are compressible or sparse in some domain due to the homogeneity, compactness and regularity of structures. Instead of sampling the entire data and then compress it to eliminate redundancy, CS performs a compressed data acquisition. Some basic requirements to enable Compressed Sensing are (i) to find a sparsifying transform able to shrink the data into a small number of coefficients (ii) to acquire random projections of the signal into orthogonal subspaces, such as the Fourier domain for spatially-sparse images (iii) to use a sampling scheme that obeys the Restricted Isometry Property (RIP) [9] and (iv) to use a sampling domain and a sparsifying transform that span incoherent domains (i.e. domains where the signal is dense in one case and sparse in the other one) [6].
Complying with these requirements, CS states that a signal g ∈ R N having a S-sparse representation (i.e. it can be well represented by a small number S of coefficients, where S ≪ N ) on a basis Ψ, can be reconstructed very accurately from a small number of projections of g onto randomly chosen subspaces (e.g. Fourier measurements for spatial sparsity). More precisely, a signal g has a sparse representation if it can be written as a linear combination of a small set of vectors taken from some basis Ψ, such as g = N i c i Ψ i , with c ℓ1 ≈ S, where • ℓ1 denotes the ℓ 1 norm which corresponds to the sum of magnitudes of all terms of the candidate signal g projected on Ψ. In general, the ℓ p norm is defined as
As demonstrated in [8], if such a sparsifying transform Ψ exists in the spatial domain, it is possible to reconstruct an image g from partial knowledge of its Fourier spectrum. In our case, g will represent the local optical intensity in the object plane. We denote f ∈ C N the associated complex optical field, satisfying g = |f | 2 . The radiation field propagates from the object to the detector plane in Fresnel diffraction conditions. Thus, the optical field in the object plane f is linked to the field F in the detection plane by a Fresnel transform, expressed in the discrete case as:
where n, p ∈ {1, . . . , N } denote pixel indexes, α ∈ R + is the parameter of the quadratic phase factor e iαn 2 describing the curvature in the detection plane of a wave emitted by a point source in the object plane. In CS, the signal reconstruction consists in solving a convex optimization problem that finds the candidate ĝ (• denotes an estimator) of minimal complexity satisfying
The experimental setup is sketched in Fig. 1. It consists of an off-axis, frequency-shifting digital holography scheme [10,11] F can be back-propagated numerically to the target plane with the standard convolution method when all measurements F ∈ C N are available. In this case, the complex field in the object plane f is retrieved from a discrete inverse Fresnel transform of F ; f = F -1 (F ) :
F n e -i(αn 2 -2πnp/N ) (
Now returning to the CS reconstruction problem, we want to recover the intensity image of the object g = {|f | 2 : f ∈ C N } from a small number of measurements F | Γ ∈ C M where M ≪ N . Partial measurements in the detection plane, illustrated by the first step in Fig. 2, can be written as F | Γ = Φf , where the sampling matrix Φ models a discrete Fresnel transform (eq. 1) and random undersampling with flat distribution. To find the best estimator ĝ, we solve the following convex optimization problem [12]:
This opt
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